fof(s14_fraenkel,theorem,(
    a_0_24_fraenkel = a_0_26_fraenkel ),
    inference(mizar_proof,[status(thm)],[s3_fraenkel]),
    [file(fraenkel,s14_fraenkel),interesting(1.00)]).

fof(s15_fraenkel,theorem,(
    a_0_27_fraenkel = a_0_29_fraenkel ),
    inference(mizar_proof,[status(thm)],[s4_fraenkel]),
    [file(fraenkel,s15_fraenkel),interesting(1.00)]).

fof(t16_fraenkel,theorem,(
    ! [A,B] : 
      ( ( v1_finset_1(A)
        & v1_finset_1(B) )
     => v1_finset_1(k1_funct_2(A,B)) ) ),
    inference(mizar_proof,[status(thm)],[t5_fraenkel,t19_finset_1,t24_finset_1,t13_finset_1]),
    [file(fraenkel,t16_fraenkel),interesting(0.95)]).

fof(t5_fraenkel,theorem,(
    ! [A,B] : r1_tarski(k1_funct_2(A,B),k1_zfmisc_1(k2_zfmisc_1(A,B))) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d2_funct_2,t21_relat_1,t118_zfmisc_1,t1_xboole_1]),
    [file(fraenkel,t5_fraenkel),interesting(0.91)]).

fof(s18_fraenkel,theorem,(
    r1_xboole_0(a_0_33_fraenkel,f2_s18_fraenkel) ),
    inference(mizar_proof,[status(thm)],[t3_xboole_0]),
    [file(fraenkel,s18_fraenkel),interesting(0.90)]).

fof(s21_fraenkel,theorem,
    ( v1_finset_1(f2_s21_fraenkel)
   => v1_finset_1(a_0_38_fraenkel) ),
    inference(mizar_proof,[status(thm)],[s3_funct_1,d3_tarski,d3_xboole_0,d12_funct_1,d10_xboole_0,d3_tarski,d12_funct_1,d3_xboole_0,t17_finset_1]),
    [file(fraenkel,s21_fraenkel),interesting(0.90)]).

fof(s20_fraenkel,theorem,(
    a_0_36_fraenkel = a_0_37_fraenkel ),
    inference(mizar_proof,[status(thm)],[s19_fraenkel]),
    [file(fraenkel,s20_fraenkel),interesting(0.86)]).

fof(s10_fraenkel,theorem,(
    r1_tarski(a_0_18_fraenkel,f1_s10_fraenkel) ),
    inference(mizar_proof,[status(thm)],[s7_domain_1]),
    [file(fraenkel,s10_fraenkel),interesting(0.79)]).

fof(s13_fraenkel,theorem,(
    a_0_21_fraenkel = a_0_23_fraenkel ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d10_xboole_0,d3_tarski]),
    [file(fraenkel,s13_fraenkel),interesting(0.78)]).

fof(s22_fraenkel,theorem,
    ( ( v1_finset_1(f3_s22_fraenkel)
      & v1_finset_1(f4_s22_fraenkel) )
   => v1_finset_1(a_0_39_fraenkel) ),
    inference(mizar_proof,[status(thm)],[t19_finset_1,s3_funct_1,t123_zfmisc_1,d3_tarski,t106_zfmisc_1,d3_xboole_0,t7_mcart_1,d12_funct_1,d10_xboole_0,d3_tarski,d12_funct_1,d3_xboole_0,t23_mcart_1,t106_zfmisc_1,t17_finset_1]),
    [file(fraenkel,s22_fraenkel),interesting(0.77)]).

fof(t15_fraenkel,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B,C,D] : 
          ( m2_fraenkel(D,B,A,k1_fraenkel(B,A))
         => k7_relat_1(D,C) = k7_relat_1(D,k3_xboole_0(B,C)) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_funct_2,t90_relat_1,t16_xboole_1,t68_funct_1,t68_funct_1]),
    [file(fraenkel,t15_fraenkel),interesting(0.77)]).

fof(t11_fraenkel,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : m1_fraenkel(k1_funct_2(B,A),B,A) ) ),
    inference(mizar_proof,[status(thm)],[t11_funct_2,t121_funct_2,d2_fraenkel]),
    [file(fraenkel,t11_fraenkel),interesting(0.77)]).

fof(s3_fraenkel,theorem,
    ( ! [A] : 
        ( m1_subset_1(A,f1_s3_fraenkel)
       => ( p1_s3_fraenkel(A)
        <=> p2_s3_fraenkel(A) ) )
   => a_0_4_fraenkel = a_0_5_fraenkel ),
    inference(mizar_proof,[status(thm)],[s1_fraenkel,d10_xboole_0,s1_fraenkel]),
    [file(fraenkel,s3_fraenkel),interesting(0.74)]).

fof(t8_fraenkel,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => v1_fraenkel(k1_tarski(A)) ) ),
    inference(mizar_proof,[status(thm)],[d1_fraenkel,d1_tarski]),
    [file(fraenkel,t8_fraenkel),interesting(0.74)]).

fof(s17_fraenkel,theorem,(
    r1_tarski(a_0_32_fraenkel,f2_s17_fraenkel) ),
    inference(mizar_proof,[status(thm)],[d3_tarski]),
    [file(fraenkel,s17_fraenkel),interesting(0.73)]).

fof(t17_fraenkel,theorem,(
    ! [A] : 
      ( v1_fraenkel(A)
     => ! [B] : 
          ( r1_tarski(B,A)
         => v1_fraenkel(B) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_fraenkel,d1_fraenkel]),
    [file(fraenkel,t17_fraenkel),interesting(0.68)]).

fof(s4_fraenkel,theorem,
    ( ! [A] : 
        ( m1_subset_1(A,f1_s4_fraenkel)
       => ! [B] : 
            ( m1_subset_1(B,f2_s4_fraenkel)
           => ( p1_s4_fraenkel(A,B)
            <=> p2_s4_fraenkel(A,B) ) ) )
   => a_0_6_fraenkel = a_0_7_fraenkel ),
    inference(mizar_proof,[status(thm)],[s2_fraenkel,d10_xboole_0,s2_fraenkel]),
    [file(fraenkel,s4_fraenkel),interesting(0.65)]).

fof(s1_fraenkel,theorem,
    ( ! [A] : 
        ( m1_subset_1(A,f1_s1_fraenkel)
       => ( p1_s1_fraenkel(A)
         => p2_s1_fraenkel(A) ) )
   => r1_tarski(a_0_0_fraenkel,a_0_1_fraenkel) ),
    inference(mizar_proof,[status(thm)],[d3_tarski]),
    [file(fraenkel,s1_fraenkel),interesting(0.58)]).

fof(t14_fraenkel,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B,C,D] : 
          ( ( r1_tarski(C,B)
            & r1_tarski(D,A) )
         => ( k1_funct_2(C,D) = k1_xboole_0
            | ! [E] : 
                ( m1_subset_1(E,k1_funct_2(C,D))
               => ? [F] : 
                    ( m2_fraenkel(F,B,A,k1_fraenkel(B,A))
                    & k7_relat_1(F,C) = E ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_funct_2,t6_fraenkel,t136_funct_2,t11_funct_2,t28_xboole_1,d1_funct_2,t68_funct_1]),
    [file(fraenkel,t14_fraenkel),interesting(0.58)]).

fof(s6_fraenkel,theorem,
    ( ! [A] : 
        ( m1_subset_1(A,f1_s6_fraenkel)
       => ( p1_s6_fraenkel(A)
         => f2_s6_fraenkel(A) = f3_s6_fraenkel(A) ) )
   => a_0_10_fraenkel = a_0_11_fraenkel ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d3_tarski,d10_xboole_0]),
    [file(fraenkel,s6_fraenkel),interesting(0.53)]).

fof(s7_fraenkel,theorem,
    ( ! [A] : 
        ( m1_subset_1(A,f1_s7_fraenkel)
       => ! [B] : 
            ( m1_subset_1(B,f2_s7_fraenkel)
           => f3_s7_fraenkel(A,B) = f4_s7_fraenkel(A,B) ) )
   => a_0_12_fraenkel = a_0_13_fraenkel ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d3_tarski,d10_xboole_0]),
    [file(fraenkel,s7_fraenkel),interesting(0.50)]).

fof(s2_fraenkel,theorem,
    ( ! [A] : 
        ( m1_subset_1(A,f1_s2_fraenkel)
       => ! [B] : 
            ( m1_subset_1(B,f2_s2_fraenkel)
           => ( p1_s2_fraenkel(A,B)
             => p2_s2_fraenkel(A,B) ) ) )
   => r1_tarski(a_0_2_fraenkel,a_0_3_fraenkel) ),
    inference(mizar_proof,[status(thm)],[d3_tarski]),
    [file(fraenkel,s2_fraenkel),interesting(0.49)]).

fof(s25_fraenkel,theorem,
    ( ( v1_finset_1(f3_s25_fraenkel)
      & v1_finset_1(f4_s25_fraenkel)
      & ! [A] : 
          ( m2_fraenkel(A,f1_s25_fraenkel,f2_s25_fraenkel,k1_fraenkel(f1_s25_fraenkel,f2_s25_fraenkel))
         => ! [B] : 
              ( m2_fraenkel(B,f1_s25_fraenkel,f2_s25_fraenkel,k1_fraenkel(f1_s25_fraenkel,f2_s25_fraenkel))
             => ( k7_relat_1(A,f3_s25_fraenkel) = k7_relat_1(B,f3_s25_fraenkel)
               => f5_s25_fraenkel(A) = f5_s25_fraenkel(B) ) ) ) )
   => v1_finset_1(a_0_40_fraenkel) ),
    inference(mizar_proof,[status(thm)],[d1_funct_2,t12_relset_1,t19_xboole_1,t3_xboole_1,t151_relat_1,d7_xboole_0,t11_funct_2,t17_xboole_1,t17_xboole_1,t14_fraenkel,t15_fraenkel,d2_funct_2,t1_xboole_1,t148_relat_1,s3_funct_2,d3_tarski,t90_relat_1,d1_funct_2,t12_relset_1,t76_funct_1,t1_xboole_1,t148_relat_1,t19_xboole_1,d2_funct_2,d2_funct_2,d1_funct_2,d12_funct_1,d10_xboole_0,t15_finset_1,t16_fraenkel,t17_finset_1]),
    [file(fraenkel,s25_fraenkel),interesting(0.45)]).

fof(s19_fraenkel,theorem,
    ( ! [A] : 
        ( m1_subset_1(A,f1_s19_fraenkel)
       => ! [B] : 
            ( m1_subset_1(B,f2_s19_fraenkel)
           => ( p2_s19_fraenkel(A,B)
            <=> ( B = f4_s19_fraenkel
                & p1_s19_fraenkel(A,B) ) ) ) )
   => a_0_34_fraenkel = a_0_35_fraenkel ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d10_xboole_0,d3_tarski]),
    [file(fraenkel,s19_fraenkel),interesting(0.44)]).

fof(s5_fraenkel,theorem,
    ( ! [A] : 
        ( m1_subset_1(A,f1_s5_fraenkel)
       => f2_s5_fraenkel(A) = f3_s5_fraenkel(A) )
   => a_0_8_fraenkel = a_0_9_fraenkel ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d3_tarski,d10_xboole_0]),
    [file(fraenkel,s5_fraenkel),interesting(0.42)]).

fof(t6_fraenkel,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B,C,D] : 
          ( ( r1_tarski(C,B)
            & r1_tarski(D,A) )
         => ( k1_funct_2(C,D) = k1_xboole_0
            | ! [E] : 
                ( m1_subset_1(E,k1_funct_2(C,D))
               => ( v1_funct_1(E)
                  & m2_relset_1(E,B,A) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_funct_2,t1_xboole_1,t11_relset_1]),
    [file(fraenkel,t6_fraenkel),interesting(0.39)]).

fof(t10_fraenkel,theorem,(
    ! [A,B,C] : 
      ( ( v1_funct_1(C)
        & v1_funct_2(C,A,B)
        & m2_relset_1(C,A,B) )
     => m1_fraenkel(k2_setwiseo(k1_zfmisc_1(k2_zfmisc_1(A,B)),C),A,B) ) ),
    inference(mizar_proof,[status(thm)],[t8_fraenkel,d1_tarski,d2_fraenkel]),
    [file(fraenkel,t10_fraenkel),interesting(0.37)]).

fof(s24_fraenkel,theorem,(
    ? [A] : 
      ( m1_subset_1(A,k5_finsub_1(f1_s24_fraenkel))
      & ! [B] : 
          ( m1_subset_1(B,f1_s24_fraenkel)
         => ( r2_hidden(B,A)
          <=> ? [C] : 
                ( m1_subset_1(C,f2_s24_fraenkel)
                & r2_hidden(C,f3_s24_fraenkel)
                & B = f4_s24_fraenkel(C)
                & p1_s24_fraenkel(B,C) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[s21_fraenkel,d3_tarski,t13_finset_1,s10_fraenkel,d5_finsub_1]),
    [file(fraenkel,s24_fraenkel),interesting(0.32)]).

fof(s8_fraenkel,theorem,
    ( ( ! [A] : 
          ( m1_subset_1(A,f1_s8_fraenkel)
         => ! [B] : 
              ( m1_subset_1(B,f2_s8_fraenkel)
             => ( p1_s8_fraenkel(A,B)
              <=> p2_s8_fraenkel(A,B) ) ) )
      & ! [A] : 
          ( m1_subset_1(A,f1_s8_fraenkel)
         => ! [B] : 
              ( m1_subset_1(B,f2_s8_fraenkel)
             => f3_s8_fraenkel(A,B) = f3_s8_fraenkel(B,A) ) ) )
   => a_0_14_fraenkel = a_0_15_fraenkel ),
    inference(mizar_proof,[status(thm)],[s2_fraenkel,s2_fraenkel,s7_fraenkel,d10_xboole_0]),
    [file(fraenkel,s8_fraenkel),interesting(0.32)]).

fof(s9_fraenkel,theorem,
    ( ( k7_relat_1(f4_s9_fraenkel,f3_s9_fraenkel) = k7_relat_1(f5_s9_fraenkel,f3_s9_fraenkel)
      & ! [A] : 
          ( m1_subset_1(A,f1_s9_fraenkel)
         => ( r2_hidden(A,f3_s9_fraenkel)
           => ( p1_s9_fraenkel(A)
            <=> p2_s9_fraenkel(A) ) ) ) )
   => a_0_16_fraenkel = a_0_17_fraenkel ),
    inference(mizar_proof,[status(thm)],[s3_fraenkel,t3_fraenkel,s6_fraenkel]),
    [file(fraenkel,s9_fraenkel),interesting(0.22)]).

fof(t3_fraenkel,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ! [B] : 
          ( ~ v1_xboole_0(B)
         => ! [C] : 
              ( ( v1_funct_1(C)
                & v1_funct_2(C,A,B)
                & m2_relset_1(C,A,B) )
             => ! [D] : 
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,A,B)
                    & m2_relset_1(D,A,B) )
                 => ! [E] : 
                      ( k7_relat_1(C,E) = k7_relat_1(D,E)
                     => ! [F] : 
                          ( m1_subset_1(F,A)
                         => ( r2_hidden(F,E)
                           => k8_funct_2(A,B,C,F) = k8_funct_2(A,B,D,F) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t72_funct_1,t72_funct_1]),
    [file(fraenkel,t3_fraenkel),interesting(0.12)]).

fof(s27_fraenkel,theorem,
    ( ! [A] : 
        ( m1_subset_1(A,f1_s27_fraenkel)
       => ~ ( r2_hidden(A,f3_s27_fraenkel)
            & ! [B] : 
                ( m1_subset_1(B,f2_s27_fraenkel)
               => ~ p1_s27_fraenkel(A,B) ) ) )
   => ? [A] : 
        ( m2_fraenkel(A,f1_s27_fraenkel,f2_s27_fraenkel,k1_fraenkel(f1_s27_fraenkel,f2_s27_fraenkel))
        & ! [B] : 
            ( m1_subset_1(B,f1_s27_fraenkel)
           => ( r2_hidden(B,f3_s27_fraenkel)
             => p1_s27_fraenkel(B,k8_funct_2(f1_s27_fraenkel,f2_s27_fraenkel,A,B)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[s26_fraenkel,t11_funct_2]),
    [file(fraenkel,s27_fraenkel),interesting(0.09)]).

fof(s26_fraenkel,theorem,
    ( ! [A] : 
        ( m1_subset_1(A,f1_s26_fraenkel)
       => ~ ( r2_hidden(A,f3_s26_fraenkel)
            & ! [B] : 
                ( m1_subset_1(B,f2_s26_fraenkel)
               => ~ p1_s26_fraenkel(A,B) ) ) )
   => ? [A] : 
        ( v1_funct_1(A)
        & v1_funct_2(A,f1_s26_fraenkel,f2_s26_fraenkel)
        & m2_relset_1(A,f1_s26_fraenkel,f2_s26_fraenkel)
        & ! [B] : 
            ( m1_subset_1(B,f1_s26_fraenkel)
           => ( r2_hidden(B,f3_s26_fraenkel)
             => p1_s26_fraenkel(B,k8_funct_2(f1_s26_fraenkel,f2_s26_fraenkel,A,B)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t28_wellord2,s3_funct_2]),
    [file(fraenkel,s26_fraenkel),interesting(0.04)]).

fof(s11_fraenkel,theorem,
    ( ! [A] : 
        ( r2_hidden(A,a_0_19_fraenkel)
       => p2_s11_fraenkel(A) )
   => ! [A] : 
        ( m1_subset_1(A,f1_s11_fraenkel)
       => ! [B] : 
            ( m1_subset_1(B,f2_s11_fraenkel)
           => ( p1_s11_fraenkel(A,B)
             => p2_s11_fraenkel(f3_s11_fraenkel(A,B)) ) ) ) ),
    file(fraenkel,s11_fraenkel),
    [interesting(0.00)]).

fof(s12_fraenkel,theorem,
    ( ! [A] : 
        ( m1_subset_1(A,f1_s12_fraenkel)
       => ! [B] : 
            ( m1_subset_1(B,f2_s12_fraenkel)
           => ( p1_s12_fraenkel(A,B)
             => p2_s12_fraenkel(f3_s12_fraenkel(A,B)) ) ) )
   => ! [A] : 
        ( r2_hidden(A,a_0_20_fraenkel)
       => p2_s12_fraenkel(A) ) ),
    file(fraenkel,s12_fraenkel),
    [interesting(0.00)]).

fof(d3_tarski,definition,(
    ! [A,B] : 
      ( r1_tarski(A,B)
    <=> ! [C] : 
          ( r2_hidden(C,A)
         => r2_hidden(C,B) ) ) ),
    file(tarski,d3_tarski),
    [interesting(0.00)]).

fof(d10_xboole_0,definition,(
    ! [A,B] : 
      ( A = B
    <=> ( r1_tarski(A,B)
        & r1_tarski(B,A) ) ) ),
    file(xboole_0,d10_xboole_0),
    [interesting(0.00)]).

fof(s16_fraenkel,theorem,
    ( ! [A] : 
        ( m1_subset_1(A,f1_s16_fraenkel)
       => ! [B] : 
            ( m1_subset_1(B,f2_s16_fraenkel)
           => ~ ( p1_s16_fraenkel(A,B)
                & ! [C] : 
                    ( m1_subset_1(C,f1_s16_fraenkel)
                   => ~ ( p2_s16_fraenkel(C,B)
                        & f3_s16_fraenkel(A,B) = f3_s16_fraenkel(C,B) ) ) ) ) )
   => r1_tarski(a_0_30_fraenkel,a_0_31_fraenkel) ),
    inference(mizar_proof,[status(thm)],[d3_tarski]),
    [file(fraenkel,s16_fraenkel),interesting(0.00)]).

fof(t3_xboole_0,theorem,(
    ! [A,B] : 
      ( ~ ( ~ r1_xboole_0(A,B)
          & ! [C] : ~ ( r2_hidden(C,A)
              & r2_hidden(C,B) ) )
      & ~ ( ? [C] : 
              ( r2_hidden(C,A)
              & r2_hidden(C,B) )
          & r1_xboole_0(A,B) ) ) ),
    file(xboole_0,t3_xboole_0),
    [interesting(0.00)]).

fof(t19_finset_1,theorem,(
    ! [A,B] : 
      ( ( v1_finset_1(A)
        & v1_finset_1(B) )
     => v1_finset_1(k2_zfmisc_1(A,B)) ) ),
    file(finset_1,t19_finset_1),
    [interesting(0.00)]).

fof(s3_funct_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & k1_relat_1(A) = f1_s3_funct_1
      & ! [B] : 
          ( r2_hidden(B,f1_s3_funct_1)
         => k1_funct_1(A,B) = f2_s3_funct_1(B) ) ) ),
    file(funct_1,s3_funct_1),
    [interesting(0.00)]).

fof(t123_zfmisc_1,theorem,(
    ! [A,B,C,D] : k2_zfmisc_1(k3_xboole_0(A,B),k3_xboole_0(C,D)) = k3_xboole_0(k2_zfmisc_1(A,C),k2_zfmisc_1(B,D)) ),
    file(zfmisc_1,t123_zfmisc_1),
    [interesting(0.00)]).

fof(t106_zfmisc_1,theorem,(
    ! [A,B,C,D] : 
      ( r2_hidden(k4_tarski(A,B),k2_zfmisc_1(C,D))
    <=> ( r2_hidden(A,C)
        & r2_hidden(B,D) ) ) ),
    file(zfmisc_1,t106_zfmisc_1),
    [interesting(0.00)]).

fof(d3_xboole_0,definition,(
    ! [A,B,C] : 
      ( C = k3_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            & r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d3_xboole_0),
    [interesting(0.00)]).

fof(t7_mcart_1,theorem,(
    ! [A,B] : 
      ( k1_mcart_1(k4_tarski(A,B)) = A
      & k2_mcart_1(k4_tarski(A,B)) = B ) ),
    file(mcart_1,t7_mcart_1),
    [interesting(0.00)]).

fof(d12_funct_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C] : 
          ( C = k9_relat_1(A,B)
        <=> ! [D] : 
              ( r2_hidden(D,C)
            <=> ? [E] : 
                  ( r2_hidden(E,k1_relat_1(A))
                  & r2_hidden(E,B)
                  & D = k1_funct_1(A,E) ) ) ) ) ),
    file(funct_1,d12_funct_1),
    [interesting(0.00)]).

fof(t23_mcart_1,theorem,(
    ! [A,B,C] : 
      ( r2_hidden(A,k2_zfmisc_1(B,C))
     => A = k4_tarski(k1_mcart_1(A),k2_mcart_1(A)) ) ),
    file(mcart_1,t23_mcart_1),
    [interesting(0.00)]).

fof(t17_finset_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( v1_finset_1(A)
       => v1_finset_1(k9_relat_1(B,A)) ) ) ),
    file(finset_1,t17_finset_1),
    [interesting(0.00)]).

fof(t6_setwiseo,theorem,(
    ! [A,B,C] : 
      ( r2_hidden(A,k2_xboole_0(B,k1_tarski(C)))
    <=> ( r2_hidden(A,B)
        | A = C ) ) ),
    file(setwiseo,t6_setwiseo),
    [interesting(0.00)]).

fof(d2_xboole_0,definition,(
    ! [A,B,C] : 
      ( C = k2_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            | r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d2_xboole_0),
    [interesting(0.00)]).

fof(s4_setwiseo,theorem,
    ( ( p1_s4_setwiseo(k1_setwiseo(f1_s4_setwiseo))
      & ! [A] : 
          ( m1_subset_1(A,k5_finsub_1(f1_s4_setwiseo))
         => ! [B] : 
              ( m1_subset_1(B,f1_s4_setwiseo)
             => ( p1_s4_setwiseo(A)
               => p1_s4_setwiseo(k5_setwiseo(f1_s4_setwiseo,A,k2_setwiseo(f1_s4_setwiseo,B))) ) ) ) )
   => ! [A] : 
        ( m1_subset_1(A,k5_finsub_1(f1_s4_setwiseo))
       => p1_s4_setwiseo(A) ) ),
    file(setwiseo,s4_setwiseo),
    [interesting(0.00)]).

fof(s23_fraenkel,theorem,
    ( ( ! [A] : 
          ( m1_subset_1(A,f1_s23_fraenkel)
         => p1_s23_fraenkel(A,A) )
      & ! [A] : 
          ( m1_subset_1(A,f1_s23_fraenkel)
         => ! [B] : 
              ( m1_subset_1(B,f1_s23_fraenkel)
             => ! [C] : 
                  ( m1_subset_1(C,f1_s23_fraenkel)
                 => ( ( p1_s23_fraenkel(A,B)
                      & p1_s23_fraenkel(B,C) )
                   => p1_s23_fraenkel(A,C) ) ) ) ) )
   => ! [A] : 
        ( m1_subset_1(A,f1_s23_fraenkel)
       => ~ ( r2_hidden(A,f2_s23_fraenkel)
            & ! [B] : 
                ( m1_subset_1(B,f1_s23_fraenkel)
               => ~ ( r2_hidden(B,f2_s23_fraenkel)
                    & p1_s23_fraenkel(B,A)
                    & ! [C] : 
                        ( m1_subset_1(C,f1_s23_fraenkel)
                       => ( ( r2_hidden(C,f2_s23_fraenkel)
                            & p1_s23_fraenkel(C,B) )
                         => p1_s23_fraenkel(B,C) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t6_setwiseo,t6_setwiseo,d2_xboole_0,t6_setwiseo,d2_xboole_0,t6_setwiseo,t6_setwiseo,t6_setwiseo,t6_setwiseo,s4_setwiseo]),
    [file(fraenkel,s23_fraenkel),interesting(0.00)]).

fof(t13_finset_1,theorem,(
    ! [A,B] : 
      ( ( r1_tarski(A,B)
        & v1_finset_1(B) )
     => v1_finset_1(A) ) ),
    file(finset_1,t13_finset_1),
    [interesting(0.00)]).

fof(s7_domain_1,theorem,(
    m1_subset_1(a_0_0_domain_1,k1_zfmisc_1(f1_s7_domain_1)) ),
    file(domain_1,s7_domain_1),
    [interesting(0.00)]).

fof(d5_finsub_1,definition,(
    ! [A,B] : 
      ( v4_finsub_1(B)
     => ( B = k5_finsub_1(A)
      <=> ! [C] : 
            ( r2_hidden(C,B)
          <=> ( r1_tarski(C,A)
              & v1_finset_1(C) ) ) ) ) ),
    file(finsub_1,d5_finsub_1),
    [interesting(0.00)]).

fof(d1_funct_2,definition,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
     => ( ( ( B = k1_xboole_0
           => A = k1_xboole_0 )
         => ( v1_funct_2(C,A,B)
          <=> A = k4_relset_1(A,B,C) ) )
        & ( B = k1_xboole_0
         => ( A = k1_xboole_0
            | ( v1_funct_2(C,A,B)
            <=> C = k1_xboole_0 ) ) ) ) ) ),
    file(funct_2,d1_funct_2),
    [interesting(0.00)]).

fof(t12_relset_1,theorem,(
    ! [A,B,C] : 
      ( m2_relset_1(C,A,B)
     => ( r1_tarski(k1_relat_1(C),A)
        & r1_tarski(k2_relat_1(C),B) ) ) ),
    file(relset_1,t12_relset_1),
    [interesting(0.00)]).

fof(t19_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(A,C) )
     => r1_tarski(A,k3_xboole_0(B,C)) ) ),
    file(xboole_1,t19_xboole_1),
    [interesting(0.00)]).

fof(t3_xboole_1,theorem,(
    ! [A] : 
      ( r1_tarski(A,k1_xboole_0)
     => A = k1_xboole_0 ) ),
    file(xboole_1,t3_xboole_1),
    [interesting(0.00)]).

fof(t151_relat_1,theorem,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => ( k9_relat_1(B,A) = k1_xboole_0
      <=> r1_xboole_0(k1_relat_1(B),A) ) ) ),
    file(relat_1,t151_relat_1),
    [interesting(0.00)]).

fof(d7_xboole_0,definition,(
    ! [A,B] : 
      ( r1_xboole_0(A,B)
    <=> k3_xboole_0(A,B) = k1_xboole_0 ) ),
    file(xboole_0,d7_xboole_0),
    [interesting(0.00)]).

fof(t11_funct_2,theorem,(
    ! [A,B,C] : 
      ( ( v1_funct_1(C)
        & v1_funct_2(C,A,B)
        & m2_relset_1(C,A,B) )
     => ( ( B = k1_xboole_0
         => A = k1_xboole_0 )
       => r2_hidden(C,k1_funct_2(A,B)) ) ) ),
    file(funct_2,t11_funct_2),
    [interesting(0.00)]).

fof(t17_xboole_1,theorem,(
    ! [A,B] : r1_tarski(k3_xboole_0(A,B),A) ),
    file(xboole_1,t17_xboole_1),
    [interesting(0.00)]).

fof(d2_funct_2,definition,(
    ! [A,B,C] : 
      ( C = k1_funct_2(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ? [E] : 
              ( v1_relat_1(E)
              & v1_funct_1(E)
              & D = E
              & k1_relat_1(E) = A
              & r1_tarski(k2_relat_1(E),B) ) ) ) ),
    file(funct_2,d2_funct_2),
    [interesting(0.00)]).

fof(t1_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(B,C) )
     => r1_tarski(A,C) ) ),
    file(xboole_1,t1_xboole_1),
    [interesting(0.00)]).

fof(t11_relset_1,theorem,(
    ! [A,B,C] : 
      ( v1_relat_1(C)
     => ( ( r1_tarski(k1_relat_1(C),A)
          & r1_tarski(k2_relat_1(C),B) )
       => m2_relset_1(C,A,B) ) ) ),
    file(relset_1,t11_relset_1),
    [interesting(0.00)]).

fof(t136_funct_2,theorem,(
    ! [A,B,C] : 
      ( ( v1_funct_1(C)
        & m2_relset_1(C,A,B) )
     => ~ ( ( B = k1_xboole_0
           => A = k1_xboole_0 )
          & ! [D] : 
              ( ( v1_funct_1(D)
                & v1_funct_2(D,A,B)
                & m2_relset_1(D,A,B) )
             => ? [E] : 
                  ( r2_hidden(E,k4_relset_1(A,B,C))
                  & k1_funct_1(D,E) != k1_funct_1(C,E) ) ) ) ) ),
    file(funct_2,t136_funct_2),
    [interesting(0.00)]).

fof(t28_xboole_1,theorem,(
    ! [A,B] : 
      ( r1_tarski(A,B)
     => k3_xboole_0(A,B) = A ) ),
    file(xboole_1,t28_xboole_1),
    [interesting(0.00)]).

fof(t68_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( B = k7_relat_1(C,A)
          <=> ( k1_relat_1(B) = k3_xboole_0(k1_relat_1(C),A)
              & ! [D] : 
                  ( r2_hidden(D,k1_relat_1(B))
                 => k1_funct_1(B,D) = k1_funct_1(C,D) ) ) ) ) ) ),
    file(funct_1,t68_funct_1),
    [interesting(0.00)]).

fof(t90_relat_1,theorem,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => k1_relat_1(k7_relat_1(B,A)) = k3_xboole_0(k1_relat_1(B),A) ) ),
    file(relat_1,t90_relat_1),
    [interesting(0.00)]).

fof(t16_xboole_1,theorem,(
    ! [A,B,C] : k3_xboole_0(k3_xboole_0(A,B),C) = k3_xboole_0(A,k3_xboole_0(B,C)) ),
    file(xboole_1,t16_xboole_1),
    [interesting(0.00)]).

fof(t148_relat_1,theorem,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => k2_relat_1(k7_relat_1(B,A)) = k9_relat_1(B,A) ) ),
    file(relat_1,t148_relat_1),
    [interesting(0.00)]).

fof(s3_funct_2,theorem,
    ( ! [A] : 
        ( m1_subset_1(A,f1_s3_funct_2)
       => ? [B] : 
            ( m1_subset_1(B,f2_s3_funct_2)
            & p1_s3_funct_2(A,B) ) )
   => ? [A] : 
        ( v1_funct_1(A)
        & v1_funct_2(A,f1_s3_funct_2,f2_s3_funct_2)
        & m2_relset_1(A,f1_s3_funct_2,f2_s3_funct_2)
        & ! [B] : 
            ( m1_subset_1(B,f1_s3_funct_2)
           => p1_s3_funct_2(B,k8_funct_2(f1_s3_funct_2,f2_s3_funct_2,A,B)) ) ) ),
    file(funct_2,s3_funct_2),
    [interesting(0.00)]).

fof(t76_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( r1_tarski(k1_relat_1(k7_relat_1(B,A)),k1_relat_1(B))
        & r1_tarski(k2_relat_1(k7_relat_1(B,A)),k2_relat_1(B)) ) ) ),
    file(funct_1,t76_funct_1),
    [interesting(0.00)]).

fof(t15_finset_1,theorem,(
    ! [A,B] : 
      ( v1_finset_1(A)
     => v1_finset_1(k3_xboole_0(A,B)) ) ),
    file(finset_1,t15_finset_1),
    [interesting(0.00)]).

fof(t21_relat_1,theorem,(
    ! [A] : 
      ( v1_relat_1(A)
     => r1_tarski(A,k2_zfmisc_1(k1_relat_1(A),k2_relat_1(A))) ) ),
    file(relat_1,t21_relat_1),
    [interesting(0.00)]).

fof(t118_zfmisc_1,theorem,(
    ! [A,B,C] : 
      ( r1_tarski(A,B)
     => ( r1_tarski(k2_zfmisc_1(A,C),k2_zfmisc_1(B,C))
        & r1_tarski(k2_zfmisc_1(C,A),k2_zfmisc_1(C,B)) ) ) ),
    file(zfmisc_1,t118_zfmisc_1),
    [interesting(0.00)]).

fof(t24_finset_1,theorem,(
    ! [A] : 
      ( v1_finset_1(A)
    <=> v1_finset_1(k1_zfmisc_1(A)) ) ),
    file(finset_1,t24_finset_1),
    [interesting(0.00)]).

fof(t28_wellord2,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ~ ( ! [B] : ~ ( r2_hidden(B,A)
              & B = k1_xboole_0 )
          & ! [B] : 
              ( ( v1_relat_1(B)
                & v1_funct_1(B) )
             => ~ ( k1_relat_1(B) = A
                  & ! [C] : 
                      ( r2_hidden(C,A)
                     => r2_hidden(k1_funct_1(B,C),C) ) ) ) ) ) ),
    file(wellord2,t28_wellord2),
    [interesting(0.00)]).

fof(t72_funct_1,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r2_hidden(B,A)
       => k1_funct_1(k7_relat_1(C,A),B) = k1_funct_1(C,B) ) ) ),
    file(funct_1,t72_funct_1),
    [interesting(0.00)]).

fof(d1_fraenkel,definition,(
    ! [A] : 
      ( v1_fraenkel(A)
    <=> ! [B] : 
          ( r2_hidden(B,A)
         => ( v1_relat_1(B)
            & v1_funct_1(B) ) ) ) ),
    file(fraenkel,d1_fraenkel),
    [interesting(0.00)]).

fof(d1_tarski,definition,(
    ! [A,B] : 
      ( B = k1_tarski(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> C = A ) ) ),
    file(tarski,d1_tarski),
    [interesting(0.00)]).

fof(d2_fraenkel,definition,(
    ! [A,B,C] : 
      ( ( ~ v1_xboole_0(C)
        & v1_fraenkel(C) )
     => ( m1_fraenkel(C,A,B)
      <=> ! [D] : 
            ( m1_subset_1(D,C)
           => ( v1_funct_1(D)
              & v1_funct_2(D,A,B)
              & m2_relset_1(D,A,B) ) ) ) ) ),
    file(fraenkel,d2_fraenkel),
    [interesting(0.00)]).

fof(t121_funct_2,theorem,(
    ! [A,B,C] : 
      ( r2_hidden(C,k1_funct_2(A,B))
     => ( v1_funct_1(C)
        & v1_funct_2(C,A,B)
        & m2_relset_1(C,A,B) ) ) ),
    file(funct_2,t121_funct_2),
    [interesting(0.00)]).

fof(t12_fraenkel,theorem,(
    $true ),
    file(fraenkel,t12_fraenkel),
    [interesting(0.00)]).

fof(t13_fraenkel,theorem,(
    $true ),
    file(fraenkel,t13_fraenkel),
    [interesting(0.00)]).

fof(t1_fraenkel,theorem,(
    $true ),
    file(fraenkel,t1_fraenkel),
    [interesting(0.00)]).

fof(t2_fraenkel,theorem,(
    $true ),
    file(fraenkel,t2_fraenkel),
    [interesting(0.00)]).

fof(t4_fraenkel,theorem,(
    $true ),
    file(fraenkel,t4_fraenkel),
    [interesting(0.00)]).

fof(t7_fraenkel,theorem,(
    $true ),
    file(fraenkel,t7_fraenkel),
    [interesting(0.00)]).

fof(t9_fraenkel,theorem,(
    $true ),
    file(fraenkel,t9_fraenkel),
    [interesting(0.00)]).